Convexity in ℝⁿ

Abstract

Let x and y be two points of ℝⁿ. The set

$$ [x,y]: = \{ z \in \mathbb{R}^n |z = \alpha x + (1 - \alpha )y,\alpha \in \mathbb{R}and0 \leqslant \alpha \leqslant 1\} $$

is called closed line segment joining x and y. One also denotes

$$ \begin{gathered} [(x,y): = [x,y]\backslash \{ y\} = \{ z \in \mathbb{R}^n |z = \alpha x + (1 - \alpha )y,\alpha \in \mathbb{R}and0 < \alpha \leqslant 1\} , \hfill \\ [(x,y: = [x,y]\backslash \{ x\} = \{ z \in \mathbb{R}^n |z = \alpha x + (1 - \alpha )y,\alpha \in \mathbb{R}and0 \leqslant \alpha < 1\} . \hfill \\ \end{gathered} $$

The set

$$ \begin{gathered} \left[ x \right.,\left. y \right): = \left[ {x,y} \right]\backslash \left\{ y \right\} = \left\{ {z \in \mathbb{R}^n \left| {z = \alpha x + \left( {1 - \alpha } \right)y,\alpha \in \mathbb{R} and 0 < \alpha \leqslant 1} \right.} \right\}, \hfill \\ = \left\{ {z \in \mathbb{R}^n \left| {z = \alpha x + \left( {1 - \alpha } \right)y, \alpha \in \mathbb{R} and 0 < \alpha < 1} \right.} \right\}. \hfill \\ \end{gathered} $$

is called open line segment joining x and y. It may also be noted ]x,y[.